Topological Properties of Solution Sets for Functional Differential Inclusions Governed by A Family of Operators

A.G. Ibrahim

Abstract: Let $r>0$ be a finite delay and $C([-r,t],E)$ be the Banach space of continuous functions from $[-r,0]$ to the Banach space $E$. In this paper we prove an existence theorem for functional differential inclusions of the form: $\dot u(t)\in A(t)\,u(t)+F(t,\tau(t)u)$ a.e. on $[0,T]$ and $u=\psi$ on $[-r,0]$, where $\{A(t)\dpt t\in [0,T]\}$ is a family of linear operators generating a continuous evolution operator $K(t,s)$, $F$ is a multifunction such that $F(t,\cdot)$ is weakly sequentially hemi-continuous and $\tau(t)\,u(s)=u(t+s)$, for all $t\in [0,T]$ and all $s\in [-r,0]$. Also, we are concerned with the topological properties of solution sets.